How to change the category in URL for posts in multiple categories?

How can I change the category slug that appears in the URL of a post that’s in multiple categories? I’d like to change the default WordPress behaviour of using the category with the lowest id.

I thought this would do it:

add_filter('post_link', 'mspc_post_link', 1, 2);
function mspc_post_link($link, $post) {
    if (strpos($link, 'bad-category-slug')) {
        $cats = get_the_category($post->ID);
        foreach ($cats as $cat) {
            if ("Bad Category Name" != $cat->cat_name) {
                $slug = $cat->category_nicename;
                break;
            }
        }
        $link = str_replace('bad-category-slug', $slug, $link);
    }
    return $link;
}

But while this kind of does work as expected, it doesn’t change the URL visible to user.